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The most important specifications that you will find listed for most power supplies are probably those that relate to its output signals. The reason for this is pretty obvious: providing the output voltages are what the power supply exists to do. Carefully check over all the output specifications for any power supply that you are considering using.
Some manufacturers will list separately the values for each of the specifications shown below. Other manufacturers may provide a table that shows all of the relevant output statistics (and sometimes, some of the electrical characteristics of each voltage level at the same time). There's really no difference, other than how the information is presented.
Output Rating (Watts): The nominal, total, maximum output of the power supply in watts. This is actually sometimes not even supplied in the specification sheet; the name of the power supply will usually have a number in it that is supposed to represent this value, and sometimes even does. :^) See the page on output ratings for more details.
Output Current Ratings (Maximum Load By Voltage): The maximum amount of current provided by the power supply at each voltage level. See the page on output ratings for more.
Minimum Current Ratings (Minimum Load Requirement By Voltage): The minimum amount of current that must be drawn by loads within the PC, for each voltage level it provides, in order for it to function properly. See the discussion of output power loading for more.
+3.3 V / +5 V Combination Maximum: The maximum amount of total power, in watts, that the power supply can provide simultaneously for the combination of the +3.3 V and +5 V signals. This is an upper limit that constrains any maximum load levels for either of the +3.3 V or +5 V signals individually. See the page on output ratings for more details on how this parameter works. Note that this limit may be in the "fine print" at the bottom of the output current rating specifications; make sure you find it. Note that this is only of relevance for supplies that provide +3.3 V power.
Peak Output: The amount of current that the voltage specified can supply for a limited amount of time. Usually this is specified only for the +12 V signal; see here. Ideally, the manufacturer will specify not only the peak output current but the amount of time the supply is rated to sustain that peak. For example, the continuous maximum for +12 V may be 10 A, the peak level 14 A, and the peak level may be sustainable for 10 seconds.
Output Voltage Range: For each output voltage, the range that the power supply guarantees its output to be within. Power supplies can't say that they will produce, for example, exactly +5.000 V. There's a range, and that's not a problem since systems are designed with this in mind. Generally speaking, the smaller the range the better, although I've never seen small differences here make a practical difference in using a supply. Either specific numbers will be provide (e.g., +4.8 V to +5.2 V) or a "+/-" percentage will be given (which would be +/- 4% to result in a range of +4.8 to +5.2 on a +5 voltage.) Also see the description of load regulation in the section on electrical characteristics.
Efficiency: What percentage of the total energy supplied to the power supply is converted to usable form by the power supply and conveyed to the PC's components. Typical numbers for PC power supplies are 60% to 85%; the other 15% to 40% is wasted as heat. Clearly, the more efficient the power supply, the better! Not only will you save electricity, you will ensure that the power supply runs cooler at the same time, making the supply's components last longer and the system work better overall. At the same time, don't give too much credence to this parameter, especially if you are comparing two units that have similar numbers (and many do.) 71% efficiency vs. 73% efficiency doesn't really translate to much difference in the real world. Efficiency is probably more important for supplies that provide a lot of power, since the percentages equate to larger numbers.
Power Good Delay: The typical time from when the power is applied to the supply, until the Power Good signal is asserted. The manufacturer may also specify a minimum and maximum time.
Auto Restart: If the power supply supports automatically restarting the system after an AC power failure, this will be mentioned in the specifications. See the last paragraph of this page for details.